The present invention relates generally to high data rate inter-satellite communications links—whether transmitting at light or infrared or millimeter wave or microwave or RF energies—where it is desirable to provide communications between satellites at the highest data rate and to relay terrestrially at the highest data rate commensurate with bandwidth restrictions and reliable communications, and more particularly to satellite systems defined in the time-frequency domain which transmit very short pulses or wave packets with a carrier but of a small number of cycles. (For representative pulse and packet signals, see FIGS. 1 and 2 and Barrett (1971-7) (See “References” List attached hereto). For time hopping coding schemes, see Barrett U.S. Pat. No. 5,610,907, incorporated herein by reference.
In many instances, communications between a group of satellites and ground stations is required which is of high data rate but (a) which requires a single transmitter-satellite of two or more messages, only one of which is received by two or more receiver-satellites or ground stations, or (b) which requires two or more transmitter-satellites of two or more messages, only one of which is received by two or more receiver-satellites or ground stations.
In other instances, a satellite is capable of providing extremely high data rate channels to a terrestrial downlink node, but the node is capable of relaying but a fraction of the data rate of the down channels. This is known in the industry as the “last mile” problem. The present invention provides a solution to that problem by providing bandwidth efficient use of the FCC permitted spectrum available to the user. Given a permitted possible time-bandwidth product available to the terrestrial node, there is a well known Shannon limit to the channel capacity of that node. Typically, that limit is not closely approached. The present invention permits a closer approach to the theoretic Shannon limit.
The Shannon limit is defined with respect to the Shannon channel capacity, C, in bits per second.             C      =              B        ⁢                                   ⁢                              log            2                    ⁡                      [                          1              +                              (                                  S                  N                                )                                      ]                                ,                   ⁢    or              C      =              B        ⁢                                   ⁢                              log            2                    ⁡                      [                          1              +                                                (                                                            E                      b                                                              N                      0                                                        )                                ⁢                                                                   ⁢                                  (                                      R                    B                                    )                                                      ]                                ,  where B is signal bandwidth, S=EbR is the signal input power constraint, N=N0B is the channel noise variance, Eb is the energy per bit, N0 is the spectral noise and R is the information rate (in its per second). The following inequality is a necessary consequence:       R    B    <            C      B        .  If the spectral bit rate, r, is defined:       r    =          R      B        ,then       r    <                            log          2                ⁡                  [                      1            +                          r              ⁢                                                E                  b                                                  N                  0                                                              ]                    ⁢                           ⁢      or                          E        b                    N        0              >                                        2            r                    -          1                r            .      The inequality is shown in FIG. 3. All digital communications systems can be described at locations lying below the curve in FIG. 3. Above the curve, reliable communications are not possible.
The present invention addresses signal pulses or packets arranged in a time hopping code (see Barrett U.S. Pat. No. 5,610,907 (1997)). A Gaussian approximation can be used to evaluate the probability of error for time hopping codes. The bit error rate (BER) is described by:       BER    =                            1          2                ⁢                  erfc          ⁡                      (                          Q                              2                                      )                              ≈                        exp          (                                    -                              Q                2                                      /            2                                    Q          ⁢                                    2              ⁢              π                                            ,where   Q  =            SNR        /    2.  In terms of the signal-to-noise (SNR) ratio this is:             SNR      =                        4          ⁢                      Q            2                          =                              P            2                                VAR            ⁡                          (                              K                -                1                            )                                            ,                   ⁢    or              Q      =              P                  2          ⁢                                    VAR              ⁡                              (                                  K                  -                  1                                )                                                          ,  where VAR is the variance of the cross-correction amplitude, i.E., The variance of the amplitude of the interference, P is signal power level and K are the number of satellite nodes in a network. In other words, the SNR ratio is the ratio of the autocorrelation peak squared to the variance of the amplitude of the interference. Thus, the SNR is directly proportional to the number of chips per code sequence. Introducing the data rate, R, as a trade variable, the relations become:       SNR    =                  P        2                    R        ×                  VAR          ⁡                      (                          1              -              K                        )                                ,          ⁢      Q    =                  P                  2          ⁢                                    R              ×                              VAR                ⁡                                  (                                      K                    -                    1                                    )                                                                        .      FIG. 4 shows the variation of the BER as a function of the Q parameter.
FIGS. 3 and 4 show the well known trades: Given a constant permitted signal bandwidth, B, a constant bit error rate, BE, can be obtained in the presence of noise or interference by (1) increasing the signal-to-noise (SNR) level, or (2) by decreasing the data rate, R, and thereby decreasing the spectral efficiency, r. If the requirement is a constant data rate, R, and if interference reduces the signal-to-noise (SNR) level, signal bandwidth, B, must be increased reducing the spectral bit rate, r, to the limit of zero at the bound:                     E        b                    N        0              ≈    SNR    ≥                  log        e            ⁢      2        =      0.69    ≈                  -        1.6            ⁢                           ⁢              dB        .            This ia a fundamental limit and the ratio of energy per bit to spectral noise cannot be less than 0.69 to achieve reliable communication in the presence of Gaussian noise. Reliable communication systems only exist for system ratios greater than 0.69.